Level (logarithmic quantity)

In the International System of Quantities, the level of a quantity is the logarithm of the ratio of the value of that quantity to a reference value of the same quantity.^[1][2] Examples are the various types of sound level: sound power level (literally, the level of the sound power, abbreviated SWL), sound exposure level (SEL), sound pressure level (SPL) and particle velocity level (SVL).^[3]

Mathematical definitions

Level

Level of a quantity Q, denoted L_Q, is defined by

$L_Q = \log_r\!\left(\frac{Q}{Q_0}\right)\!,$

where

• r is the base of the logarithm;
• Q is the root-power quantity;
• Q₀ is the reference value of Q.

Level of a field quantity

Level of a field quantity, denoted L_F, is defined by

$L_F = \ln\!\left(\frac{F}{F_0}\right)\!,$

where

• F is the field quantity;
• F₀ is the reference value of F.

For the level of a field quantity, the base of the logarithm is r = e.

Level of a root-power quantity

A root-power quantity is a field quantity. The Level of a root-power quantity, denoted L_F, is therefore

$L_F = \ln\!\left(\frac{F}{F_0}\right)\!,$

where

• F is the root-power quantity;
• F₀ is the reference value of F.

For the level of a root-power quantity, the base of the logarithm is r = e.

Level of a power quantity

Level of a power quantity, denoted L_P, is defined by

$L_P = \log_{e^2}\!\left(\frac{P}{P_0}\right) = \frac{1}{2} \ln\!\left(\frac{P}{P_0}\right)\!,$

where

• P is the power quantity;
• P₀ is the reference value of P.

For the level of a power quantity, the base of the logarithm is r = e².^[4]

Units of level

Power level

The neper, bel, and decibel (one tenth of a bel) are units of level that are often applied to such quantities as power, intensity, or gain. The neper, bel, and decibel are defined by

• Np = 1;
• B = (1/2) ln(10) Np;
• dB = 0.1 B = (1/20) ln(10) Np.

If F is a root-power quantity:

$L_F = \ln\!\left(\frac{F}{F_0}\right)\!~\mathrm{Np} = 2 \log_{10}\!\left(\frac{F}{F_0}\right)\!~\mathrm{B} = 20 \log_{10}\!\left(\frac{F}{F_0}\right)\!~\mathrm{dB}.$

If P is a power quantity:

$L_P = \frac{1}{2} \ln\!\left(\frac{P}{P_0}\right)\!~\mathrm{Np} = \log_{10}\!\left(\frac{P}{P_0}\right)\!~\mathrm{B} = 10 \log_{10}\!\left(\frac{P}{P_0}\right)\!~\mathrm{dB}.$

If the power quantity P is equal to F², and if the reference value of the power quantity, P₀, is equal to F₀², the levels L_F and L_P are equal.

Frequency level

The octave is a unit of level (specifically "frequency level", for r = 2) though that concept is seldom seen outside of the standard.^[5] A semitone is one twelfth of an octave.

Standardization

The level and its units are defined in ISO 80000-3.

See also

• Decibel#Definition
• Field, power, and root-power quantities
• Logarithmic scale
• Sound level

References

1. ISO 80000-3:2006, Quantities and units, Part 2: Space and Time
2. W. M. Carey, Sound Sources and Levels in the Ocean, IEEE J Oceanic Eng 31:61-75(2006)
3. ISO 80000-8:2007, Quantities and units, Part 8: Acoustics
4. Ainslie, M. A. A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound. Acoustics Today, 23 February 2015
5. ANSI/ASA S1.1-2013, Acoustical Terminology